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Stochastic-GEP-Benders.gms
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Stochastic-GEP-Benders.gms
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*-------------------------------------------------------------------------
* Universidad Pontificia Comillas de Madrid
* Optimization Techniques
* Diego Alejandro Tejada Arango
*-------------------------------------------------------------------------
$TITLE Two-Stage Stochastic Generation Expansion Planning - Using Benders
* ========================================================================
* SETS DEFINITION
* ========================================================================
SETS
p "time periods (e.g., hours) " /h01 *h24 /
sc "uncertainty scenarios " /sc01,sc02,sc03/
g "generation technologies" / wind, solar, ccgt, ocgt /
r(g) "subset of renewable technologies" / wind, solar/
l "iterations " / l01 * l50 /
ll(l) "iterations subset "
* dinamic set (to be defined depending on the input data)
sca(sc) "active scenarios"
;
* ========================================================================
* PARAMETERS AND SCALARS
* ========================================================================
SCALARS
pWeight "weight of representative period [days]" /365/
pENSCost "energy not supplied cost [kEUR/MWh]" /0.180/
;
PARAMETER
pScProb(sc) "scenario probability [p.u.]"
/sc01 0.2, sc02 0.5, sc03 0.3/
pDemand(p) "demand per time period [MW]"
/
h01 950
h02 870
h03 814
h04 779
h05 758
h06 751
h07 779
h08 834
h09 902
h10 956
h11 1010
h12 1023
h13 1018
h14 1010
h15 980
h16 965
h17 963
h18 997
h19 1093
h20 1114
h21 1115
h22 1107
h23 1053
h24 1035
/
;
TABLE pGenInfo(g,*) "generation information"
* kEUR/MWh kEUR/MW/year MW
VarCost InvCost UnitCap
ocgt 0.070 25 100
ccgt 0.050 40 400
wind 0.001 70 50
solar 0.000 50 10
;
TABLE pRenProf(p,r,sc) "renewable profile [p.u.]"
* sc01 -> low wind, high solar; sc02 -> avg wind and solar; sc03 -> high wind and low solar
wind.sc01 wind.sc02 wind.sc03 solar.sc01 solar.sc02 solar.sc03
h01 0.11 0.54 0.68 0.00 0.00 0.00
h02 0.11 0.54 0.69 0.00 0.00 0.00
h03 0.11 0.53 0.70 0.00 0.00 0.00
h04 0.11 0.52 0.71 0.00 0.00 0.00
h05 0.10 0.51 0.73 0.00 0.00 0.00
h06 0.10 0.50 0.74 0.02 0.00 0.00
h07 0.10 0.48 0.75 0.12 0.01 0.00
h08 0.09 0.47 0.76 0.30 0.07 0.01
h09 0.09 0.46 0.77 0.50 0.20 0.12
h10 0.09 0.45 0.78 0.66 0.36 0.28
h11 0.09 0.45 0.79 0.78 0.50 0.42
h12 0.09 0.45 0.80 0.83 0.57 0.51
h13 0.10 0.43 0.81 0.83 0.59 0.53
h14 0.12 0.41 0.81 0.78 0.54 0.50
h15 0.14 0.38 0.80 0.68 0.44 0.40
h16 0.15 0.35 0.79 0.53 0.29 0.23
h17 0.16 0.34 0.78 0.35 0.13 0.05
h18 0.16 0.35 0.77 0.17 0.03 0.00
h19 0.16 0.36 0.76 0.04 0.00 0.00
h20 0.15 0.38 0.75 0.00 0.00 0.00
h21 0.14 0.41 0.74 0.00 0.00 0.00
h22 0.13 0.43 0.74 0.00 0.00 0.00
h23 0.12 0.46 0.74 0.00 0.00 0.00
h24 0.12 0.48 0.74 0.00 0.00 0.00
;
* parameters for Benders decomposition
PARAMETERS
pBdTol "relative Benders' tolerance" / 1e-6 /
pZ_Lower "lower bound " / -INF /
pZ_Upper "upper bound " / INF /
pZ_Bounds_L (l,* ) "bounds as each iteration "
pInstalUnits_L (l, g ) "first stage variables values in iteration l"
pDualRenProd_L (l,p,g,sc) "dual variables of second stage constraints in iteration l"
pDualMaxProd_L (l,p,g,sc) "dual variables of second stage constraints in iteration l"
pDelta (l ) "cut type (feasibility 0 optimality 1) in iteration l"
pZ2_L (l ) "subproblem objective function value in iteration l"
;
* ========================================================================
* VARIABLES
* ========================================================================
INTEGER VARIABLE
vInstalUnits(g) "number of installed generation units [N]"
;
POSITIVE VARIABLE
vProduct(p,g,sc) "generation production per scenario [MW]"
vENS (p, sc) "energy not supplied per scenario [MW]"
;
FREE VARIABLES
* complete problem variables
vTotalCost "Total Cost = Investment + Operation [kEUR]"
vInvesCost "Total investment Cost [kEUR]"
vOperaCost "Total operating Cost [kEUR]"
* variables for Benders decomposition
vZ1 "first stage objective function "
vZ2 "second stage objective function "
vTheta "recourse function "
;
* ========================================================================
* EQUATIONS AND MODEL DEFINITION
* ========================================================================
EQUATIONS
* Complete problem equations
eTotalCost "Total Cost = Investment + Operation [kEUR]"
eInvesCost "Total investment Cost [kEUR]"
eOperaCost "Total operating Cost [kEUR]"
eBalance (p, sc) "power balance constriant [MW] "
eRenProd (p,g,sc) "renewable production constriant [MW] "
eMaxProd (p,g,sc) "generation production constraint [MW] "
* equations for Benders decomposition
eZ1 "first stage objective function "
eZ2 "second stage objective function "
eBdCuts (l ) "Benders cuts at iteration l "
;
eZ1.. vZ1 =E= vInvesCost + vTheta
;
eZ2.. vZ2 =E= + vOperaCost
;
eTotalCost.. vTotalCost =E= vInvesCost + vOperaCost
;
eInvesCost.. vInvesCost =E= SUM[g, pGenInfo(g,'InvCost')*pGenInfo(g,'UnitCap')*vInstalUnits(g)]
;
eOperaCost.. vOperaCost =E= pWeight * SUM[(p,g,sca),
pScProb(sca)*[
+ pGenInfo(g,'VarCost')*vProduct(p,g,sca)
+ pENSCost *vENS (p, sca)]]
;
eBalance(p,sca(sc))..
SUM[g,vProduct(p,g,sc)] + vENS(p,sc) =E= pDemand(p)
;
eRenProd(p,r,sca(sc))..
vProduct(p,r,sc) =L= pRenProf(p,r,sc) * pGenInfo(r,'UnitCap')*vInstalUnits(r)
;
eMaxProd(p,g,sca(sc))$[NOT r(g)]..
vProduct(p,g,sc) =L= pGenInfo(g,'UnitCap')*vInstalUnits(g)
;
eBdCuts(ll)..
pDelta(ll)* vTheta
=G= + pZ2_L(ll)
- SUM[(p,r,sc) , pDualRenProd_L(ll,p,r,sc) * pRenProf(p,r,sc) * pGenInfo(r,'UnitCap') * [pInstalUnits_L(ll,r) - vInstalUnits(r)]]
- SUM[(p,g,sc)$[NOT r(g)], pDualMaxProd_L(ll,p,g,sc) * pGenInfo(g,'UnitCap') * [pInstalUnits_L(ll,g) - vInstalUnits(g)]]
;
MODEL Master_Bd / eZ1 , eInvesCost, eBdCuts / ;
MODEL Subproblem_Bd / eZ2 , eOperaCost, eBalance, eRenProd, eMaxProd/ ;
MODEL Complete / eTotalCost, eInvesCost, eOperaCost, eBalance, eRenProd, eMaxProd/ ;
* ========================================================================
* OPTIONS AND INITIAL VALUES
* ========================================================================
* to allow CPLEX correctly detect rays in an infeasible problem
* only simplex method can be used and no preprocessing neither scaling options
* optimality and feasibility tolerances are very small to avoid primal degeneration
FILE COPT / cplex.opt /
;
PUT COPT PUTCLOSE 'ScaInd -1' / 'LPMethod 1' / 'PreInd 0' / 'EpOpt 1e-9' / 'EpRHS 1e-9' / ;
;
Subproblem_Bd.OptFile = 1 ;
;
* parameters initialization
vTheta.fx = 0 ;
LL (l ) = NO;
pInstalUnits_L (l, g ) = 0 ;
pDualRenProd_L (l,p,g,sc) = 0 ;
pDualMaxProd_L (l,p,g,sc) = 0 ;
pDelta (l ) = 0 ;
pZ2_L (l ) = 0 ;
* option to find the solution to optimality
OPTION optcr=0;
* active uncertainty scenarios with probability
sca(sc) $[pScProb(sc)] = YES ;
* ========================================================================
* BENDERS DECOMPOSITION
* ========================================================================
LOOP(l $[ABS(1-pZ_Lower/pZ_Upper) > pBdTol],
* solving master problem
solve Master_Bd using MIP minimizing vZ1 ;
* storing the master solution
pInstalUnits_L(l,g) = vInstalUnits.L(g) ;
* fixing first-stage variables and solving subproblem
vInstalUnits.FX (g) = vInstalUnits.L(g) ;
* solving subproblem
solve Subproblem_Bd using RMIP minimizing vZ2 ;
* storing parameters to build a new Benders' cut
if (Subproblem_Bd.ModelStat = 4,
pDelta(l) = 0 ;
pZ2_L (l) = Subproblem_Bd.SumInfes ;
else
* updating lower and upper bound
pZ_Lower = vZ1.L ;
pZ_Upper = MIN(pZ_Upper,vZ1.L - vTheta.L + vZ2.L) ;
pDelta(l) = 1 ;
pZ2_L (l) = vZ2.L ;
) ;
vTheta.lo = -INF ;
vTheta.up = INF ;
pDualRenProd_L (l,p,g,sc) = eRenProd.M(p,g,sc);
pDualMaxProd_L (l,p,g,sc) = eMaxProd.M(p,g,sc);
* it is important to put bounds on the master's variables
* to avoid an unbounded master problem. If the variables
* do not have initial bounds, one can impose maximum bounds
* that makes sense to the problem. Unfixing the variables
* is also necessary to let GAMS optimize againg.
vInstalUnits.LO(g) = 0 ;
* Naive approach (all to a big-number 9999)
* vInstalUnits.UP(g) = 9999 ;
* Using the problems inputs
vInstalUnits.UP(g) $[NOT pGenInfo(g,'UnitCap')]= 0 ;
vInstalUnits.UP(g) $[ pGenInfo(g,'UnitCap') AND NOT r(g)]= CEIL[SMAX[ p ,pDemand(p)/ pGenInfo(g,'UnitCap') ]];
vInstalUnits.UP(r) $[ pGenInfo(r,'UnitCap') ]= CEIL[SMAX[(p,sc),pDemand(p)/[pGenInfo(r,'UnitCap')*pRenProf(p,r,sc)+1]]];
* increase the set of Benders' cuts
LL(l) = YES ;
* save the bounds of the next iteration (to report results)
pZ_Bounds_L(l+1,'lower') = pZ_Lower ;
pZ_Bounds_L(l+1,'upper') = pZ_Upper ;
) ;
* result parameters
PARAMETERS
pInstalCap(g ) "installed capacity [MW] "
pScPrices (p,sc) "scenario prices [EUR/MWh]"
pEVPrices (p ) "expected value of prices [EUR/MWh]"
;
pInstalCap(g) = pGenInfo(g,'UnitCap')*vInstalUnits.L(g)
;
pScPrices (p,sca(sc)) = eBalance.M(p,sc) *1e3 / [pWeight * pScProb(sc)];
pEVPrices (p ) = SUM[sc, pScProb(sc) * pScPrices (p,sc)] ;
* gdx with all results
execute_unload 'TwoStageStochGEP-Benders.gdx'
*$stop
* ========================================================================
* COMPLETE MODEL SOLUTION FOR VALITATION
* ========================================================================
SOLVE Complete USING MIP MINIMIZING vTotalCost
;
* result parameters
pInstalCap(g) = pGenInfo(g,'UnitCap')*vInstalUnits.L(g)
;
pScPrices (p,sca(sc)) = eBalance.M(p,sc) *1e3 / [pWeight * pScProb(sc)];
pEVPrices (p ) = SUM[sc, pScProb(sc) * pScPrices (p,sc)] ;
* gdx with all results
execute_unload 'TwoStageStochGEP-Complete.gdx'